Dynamical Systems in Neuroscience:

Conductance-Based Models 159currentholding voltage, (mV)250inward outward0-250I (V)I 0 (V)0V-50E K-1000 10time, msa1000500currentslow conductance, g01086I-V relationsI (V)I 0 (V)b{I+I (V)-I 0 (V)}/(V-E K )V-nullclineE K 0{I-I 0 (V)}/(V-E K )42-100 -80 -60 -40 -20membrane voltage, V (mV)0 20cg-nullclineFigure 5.22: Voltage-clamp protocol to measure instantaneous (peak) and steadystate current-voltage (I-V) relations. (Shown simulations of the I Na,p +I K -model fromFig. 5.4b.)Let us illustrate the relationship using the I Na,p +I K -model, which we write in theformwhereC ˙V = I − I 0 (V ) − g(V − E K ) , (5.3)ġ = f(V, g) , (5.4)I 0 (V ) = g L (V −E L ) + g Na m ∞ (V )(V − E Na )is the instantaneous (peak) current, and g = g K n is the slow conductance. The functionf(V, g) describes the dynamics of g, and its form is not important here. The methoddescribed below is quite general, and it can be used in many circumstances when littleis known about the neuron’s electrophysiology.In Fig. 5.22 we describe a typical voltage-clamp experiment to measure the instantaneous(peak) and the steady-state I-V relations, denoted here as I 0 (V ) and I ∞ (V ),respectively. The holding voltage (Fig. 5.22a, bottom) is kept at E K and then steppedto various values V . The recorded current (Fig. 5.22a, top) typically consists of a fast(peak) component I 0 (V ) that is due to the instantaneous activation of Na + currents,leak current, and other fast currents, and then it relaxes to the asymptotic steady statevalue I ∞ (V ). Repeating this experiment for various V , one can measure the I-V functionsI 0 (V ) and I ∞ (V ) depicted in Fig. 5.22b. Notice that I 0 (V ) has the N-shape witha large region of negative slope. This region corresponds to the regenerative activation